The integer-to-integer discrete cosine and other unitary transforms become popular in recent years in such applications as lossless image coding, mobile computing, filter banks, and other areas. In this paper, we present new matrix representations of the reversible integer discrete cosine transforms (IDCT) that are based on the canonical representation and floor function. A new concept of the kernel integer discrete cosine transform is introduced, that allows us to reduce the calculation of the IDCT of type II to the kernel IDCT with a fewer operations of multiplication and floor function. The application of the kernel IDCT is described for calculation of the eight-point IDCT of type II, when seven multiplications and seven floor functions can be saved. The parameterized two-point DCT of type IV and its particular case that requires two operations of multiplication, four additions, and two floor functions are presented. The golden two-point DCT that minimizes the error of the cosine transform approximation by the IDCT is also considered. Application of the kernel DCT for calculating the eight-point IDCT results in the saving of twelve multiplications and twelve floor functions, when considered the decomposition of the transform by the Walsh-Hadamard transform.